Your problem lies in that you don't want the Optionmenu itself scrolled -- you want the drop-down which it shows to be scrolled. Unfortunatly, what Tk thinks you're trying to do is the former. It's also not in your favor that Optionmenus drop down Menus -- which, if I'm remembering correctly, can't be scrolled in the first place.

Luckily, you're not the first person to want to have a large pop-up list. Look at the BrowseEntry widget, which is automatically scrolled.

Update: Here's some code which should do more of less what you want:

my $day_menu = $date_frame->BroseEntry(
                                       -state    => "readonly",
                                       -choices  => [(1..$days{$month}
+)],
                                       -variable => \$day_number
                                      )->pack(
                                              -side=> "left"
                                             );
[download]

perl -pe '"I lo*`+$^X$\"$]!$/"=~m%(.*)%s;$_=$1;y^`+*^e v^#$&V"+@( NO CARRIER'